A key limitation of set theory is that for some properties of sets,
there is no set of all sets that satisfy the property. To address this
limitation, we consider higher order set theory, but immediately hit a
roadblock: If V contains all sets, then we cannot form structures above V.
The solution is to build higher-order set theory inside V. That is, we
can use a cardinal kappa to represent Ord (the class of all ordinals),
elements of V_kappa represent sets, subsets of V_kappa -- classes,
elements of P(P(V_kappa)) -- collections of classes, and so on.
However, not all cardinals are suitable for this purpose.
Definition: A cardinal kappa is reflective if it is correct about
higher order set theory with parameters in V_kappa.
Note: See below for an alternative equivalent definition that uses
reflection properties instead of higher order set theory.
This definition cannot be formalized in the language of set theory.
Instead, we
1. Extend the language of set theory with a predicate R: R(kappa) <==>
kappa is reflective.
2. Axiomatize the resulting extension.
3. Argue that the extension is well-defined, or at least has a solid
conceptual basis.
Let us start axiomatizing.
A1. ZFC
A2. Axiom schema of replacement for formulas involving R.
A3. R(kappa) ==> kappa is an ordinal
Now, while we cannot just formulate correctness for higher order set
theory as an axiom, the key observation is that if both kappa and lambda
are correct, then they agree with each other.
A4. (schema, phi has two free variables and does not use R)
R(kappa) and R(lambda) ==> forall s in V_min(kappa,lambda) ( phi(s,
kappa) <==> phi(s, lambda) ).
Finally, to use reflective cardinals for higher order set theory, we need:
A5. There is a proper class of reflective cardinals
Theorem: A1-A5 is equiconsistent with ZFC + Ord is subtle.
The axioms for reflective cardinals naturally correspond to the large
cardinal property of full indescribability:
A6. Schema (phi has two free variables and does not use R): If kappa
is reflective and A is a set, then
phi(kappa, A intersect kappa) ==> thereis lambda<kappa phi(lambda, A
intersect lambda)
A6 implies that reflective cardinals are strongly unfoldable (==>
totally indescribable ==> weakly compact ==> Mahlo ==> inaccessible).
Theorem: A1-A5 implies that A6 holds in HOD. A1-A6 is Pi^V_2
conservative over A1-A5.
In our presentation so far, there is still incompleteness about how
similar elements of R have to be to each other. While one option would
be to keep R open-ended and progressively reach higher expressive power
through stronger indiscernability requirements on elements of R, we
propose to make R definite by requiring R(kappa) <==> (R union {kappa}
satisfies A4). This can be formalized into a single statement, which
however is slightly technical:
A4a. forall a R(a) ==> Ord(a) (that is a is an ordinal); forall a,b,c
(Ord(a) and R(b) and R(c) and 0<a<b<c ==> (R(a) <==> forall 'phi' forall
s in V_a ( phi(a, s) holds in V_b iff phi(b, s) holds in V_c ))), where
'phi' ranges over (codes for) formulas in set theory (without R) with
two free variables.
Theorem: ZFC + A4a + A5 + "forall s (R intersect s exists)" is finitely
axiomatizable and implies A4.
A4a slightly increases the consistency strength, which, however, remains
below subtle cardinals. A consequence of A4a is that R is definable
from every proper class S subclass R. Analogously to A4a, we can
convert A6 into a single statement (which inherently makes it slightly
stronger) by using a reflective cardinal in place of V.
It remains to show that R is well-defined, or at least intuitively
sound. While V is poorly understood, the constructible universe L is a
well-understood model of set theory.
Theorem (ZFC + zero sharp): There is unique R such that (L, in, R)
satisfies A4a and R holds for a proper class of cardinals (that is
cardinals in V). (L, in, R) also satisfies A1-A6.
Moreover, we get the same theorem for other canonical inner models,
which suggests that there is unique natural way to choose R in V to
satisfy the axioms, which we intuitively describe as follows.
Key to infinitary set theory is the concept of a reflection property.
Examples of reflection properties abound -- "kappa is a cardinal",
"kappa is inaccessible", "kappa is a Sigma-2 elementary substructure of
V", and so on, and they appear to form a directed system.
Convergence Hypothesis: If ordinals a and b have sufficiently strong
reflection properties, then they satisfy the same set of statements,
even with parameters in V_min(a,b).
Definition (assuming convergence hypothesis): kappa is a reflective
cardinal, denoted by R(kappa), iff (V, kappa, in) has the same theory
with parameters in V_kappa as (V, lambda, in) for every cardinal lambda
> kappa with sufficiently strong reflection properties.
Although the notion of a reflection property is vague, the convergence
hypothesis (combined with the axiomatization) allows us to escape the
vagueness, and make the notion of R unambiguous. While our axioms are
incomplete, they are not significantly more incomplete than the axioms
of set theory. Just like ZFC, the theory can be extended with large
cardinal axioms. There are natural ways to incorporate large cardinal
notions at the full expressive level of R, and this sometimes leads to
stronger large cardinal notions.
The results -- and much more -- are in my paper:
http://web.mit.edu/dmytro/www/ReflectiveCardinals.htm
(also available on arXiv: http://arxiv.org/abs/1203.2270)
As always, I am looking for feedback, whether or not you agree with me.
Sincerely,
Dmytro Taranovsky